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Fractional-order diffusion model for multiplicative noise removal in texture-rich images and its fast explicit diffusion solving

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Abstract

In this paper, we propose a fractional-order anisotropic diffusion model based on fractional Fick’s law for multiplicative noise removal in texture-rich images. A fast explicit diffusion solving algorithm is considered. The new model is different from the model derived from the fractional-order variation method and has a clear physical background. Numerically, we use the discrete Grünwald–Letnikov approximation to implement the finite difference discretization of the model, which yields a dense coefficient matrix. To keep the method computationally tractable, limited nodes are employed in Grünwald–Letnikov approximation to produce a sparse coefficient matrix. The fast explicit diffusion method is used to speed up the calculation. The superior performance of the proposed fractional-order diffusion model is illustrated by comparing it with other denoising models on various images.

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References

  1. Argenti, F., Lapini, A., Bianchi, T.: A tutorial on speckle reduction in synthetic aperture radar images. IEEE Geosci. Remote Sens. 1(3), 6–35 (2013)

    Article  Google Scholar 

  2. Arsenault, H.H.: Speckle suppression and analysis for synthetic aperture radar images. Opt. Eng. 25(5), 636–643 (1985)

    Google Scholar 

  3. Aubert, G., Aujol, J.F.: A variational approach to removing multiplicative noise. SIAM J. Appl. Math. 68(4), 925–946 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bai, J., Feng, X.C.: Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16(10), 2492–2502 (2007)

    Article  MathSciNet  Google Scholar 

  5. Bai, J., Feng, X.C.: Image decomposition and denoising using fractional-order partial differential equations. IET Image Process. (2020). https://doi.org/10.1049/iet-ipr.2018.5499

    Article  Google Scholar 

  6. Bailey, D.L., Townsend, D.W., Valk, P.E., Maisey, M.N.: Positron Emission Tomography. Springer, London (2005)

    Book  Google Scholar 

  7. Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection–dispersion equation. Water Resour. Res. 36(6), 1403–1412 (2000)

    Article  Google Scholar 

  8. Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chan, R.H., Lanza, A., Morigi, S., Sgallari, F.: An adaptive strategy for the restoration of textured images using fractional order regularization. Numer. Math. Theory Methods Appl. 6(1), 276–296 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Che, J., Guan, Q., Wang, X.: Image denoising based on adaptive fractional partial differential equations. In: 2013 6th International Congress on Image and Signal Processing (CISP), vol. 01, pp. 288–292 (2013). https://doi.org/10.1109/CISP.2013.6744004

  11. Chen, D., Cheng, L.: Spatially adapted total variation model to remove multiplicative noise. IEEE Signal Process. Lett. 21(4), 1650–1662 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Chen, D., Sun, S., Zhang, C., Chen, Y., Xue, D.: Fractional-order \(TV-L2\) model for image denoising. Open Phys. 11(10), 1414–1422 (2013)

    Article  Google Scholar 

  13. Chen, H., Lv, W., Zhang, T.: A Kronecker product splitting preconditioner for two-dimensional space-fractional diffusion equations. J. Comput. Phys. 360, 1–14 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, Q., Montesinos, P., Sun, Q.S., Heng, P.A., Xia, D.S.: Adaptive total variation denoising based on difference curvature. Image Vis. Comput. 28(3), 298–306 (2010)

    Article  Google Scholar 

  15. Chen, S., Liu, F., Jiang, X., Turner, I., Burrage, K.: Fast finite difference approximation for identifying parameters in a two-dimensional space-fractional nonlocal model with variable diffusivity coefficients. SIAM J. Numer. Anal. 54(2), 606–624 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chen, S., Liu, F., Turner, I., Anh, V.: An implicit numerical method for the two-dimensional fractional percolation equation. Appl. Math. Comput. 219, 4322–4331 (2013)

    MathSciNet  MATH  Google Scholar 

  17. Chen, Y., Feng, W., Ranftl, R., Qiao, H., Pock, T.: A higher-order MRF based variational model for multiplicative noise reduction. IEEE Signal Process. Lett. 21(11), 1370–1374 (2014)

    Article  Google Scholar 

  18. Cuesta-Montero, E., Finat, J.: Image processing by means of a linear integro-differential equation. In: Proceedings of 3rd IASTED International Conference on Visualization, Imaging, and Image Processing, vol. 1 (2003)

  19. Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3-d transform-domain collaborative filtering. IEEE Trans. Image Process. 16(8), 2080–2095 (2007)

    Article  MathSciNet  Google Scholar 

  20. Deledalle, C.A., Denis, L., Tabti, S., Tupin, F., Reigber, A., Jager, M.: Mulog, or how to apply Gaussian denoisers to multi-channel SAR speckle reduction? IEEE Trans. Image Process. 26(9), 4389–4403 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Deledalle, C.A., Denis, L., Tupin, F.: Iterative weighted maximum likelihood denoising with probabilistic patch-based weights. IEEE Trans. Image Process. 18(12), 2661–2672 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Duits, R., Felsberg, M., Florack, L., Platel, B.: \(\alpha \) Scale spaces on a bounded domain. In: Proceedings of 4th International Conference on Scale Spaces, pp. 494–510 (2003)

  23. Gwosdek, P., Zimmer, H., Grewenig, S., Bruhn, A., Weickert, J.: A highly efficient GPU implementation for variational optic flow based on the Euler–Lagrange framework. Trends Top. Comput. Vis. 6554, 372–383 (2012)

    Google Scholar 

  24. He, J.H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167, 57–68 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hoffmann, S., Mainberger, M., Weickert, J., Puhl, M.: Compression of Depth Maps with Segment-Based Homogeneous Diffusion, vol. 7893, pp. 319–330. Springer, Berlin (2013)

    Google Scholar 

  26. Janev, M., Pilipović, S., Atanacković, T., Obradović, R., Ralević, N.: Fully fractional anisotropic diffusion for image denoising. Math. Comput. Model. 54(1), 729–741 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Krissian, K., Westin, C.F., Kikinis, R., Vosburgh, K.G.: Oriented speckle reducing anisotropic diffusion. IEEE Trans. Image Process. 16(5), 1412–1424 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lysaker, M., Lundervold, A., Xue-Cheng, T.: Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Trans. Image Process. 12(12), 1579–1590 (2003)

    Article  MATH  Google Scholar 

  29. Magin, R.L., Abdullah, O., Baleanu, D., Zhou, X.J.: Anomalous diffusion expressed through fractional order differential operators in the Bloch–Torrey equation. J. Magn. Reson. 190(2), 255–270 (2008)

    Article  Google Scholar 

  30. Meerschaert, M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56(1), 80–90 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Meerschaert, M.M., Sikorskii, A.: Fractional Calculus and Applied Analysis. De Gruyter, Berlin (2012)

    MATH  Google Scholar 

  32. Noble, J.A., Boukerroui, D.: Ultrasound image segmentation: a survey. IEEE Trans. Med. Imaging. 25(8), 987–1010 (2006)

    Article  Google Scholar 

  33. Pan, M., Zheng, L., Liu, F., Liu, C., Chen, X.: A spatial-fractional thermal transport model for nanofluid in porous media. Appl. Math. Model. 53, 622–634 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)

    Article  Google Scholar 

  35. Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. In: Math. Sci. Eng. 198, 1–40 (1998)

  36. Pu, Y.F., Zhou, J., Yuan, X.: Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement. IEEE Trans. Image Process. 19(2), 491–511 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Pu, Y.F., Zhou, J., Zhang, Y., Zhang, N., Huang, G., Siarry, P.: Fractional extreme value adaptive training method: fractional steepest descent approach. IEEE Trans. Neural Netw. Learn. Syst. 26(4), 653–662 (2015)

    Article  MathSciNet  Google Scholar 

  38. Reichel, L.: Newton interpolation at Leja points. BIT Numer. Math. 30, 332–346 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  39. Romero, P.D., Candela, V.F.: Blind deconvolution models regularized by fractional powers of the Laplacian. J. Math. Imaging Vis. 32(2), 181–191 (2008)

    Article  MathSciNet  Google Scholar 

  40. Rudin, L., Lions, P.L., Osher, S.: Multiplicative denoising and deblurring: theory and algorithms. In: Geometric Level Set Methods in Imaging, Vision, and Graphics. Springer, New York, pp. 103–119 (2003)

  41. Scalas, E., Gorenflo, R., Mainardi, F.: Application of a fractional advection–dispersion equation. Phys. A Stat. Mech. Appl. 284, 376–384 (2000)

    Article  Google Scholar 

  42. Schmidt-Richberg, A., Ehrhardt, J., Werner, R., Handels, H.: Fast Explicit Diffusion for Registration with Direction-Dependent Regularization, vol. 7359, pp. 220–228. Springer, Berlin (2012)

    Google Scholar 

  43. Schumer, R., Benson, D., Meerschaert, M., Wheatcraft, S.: Eulerian derivation of the fractional advection–dispersion equation. J. Contam. Hydrol. 48(1), 69–88 (2001)

    Article  Google Scholar 

  44. Shan, X., Sun, J., Guo, Z.: Multiplicative noise removal based on the smooth diffusion equation. J. Math. Imaging Vis. 61, 763–779 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  45. Tadjeran, C., Meerschaert, M.: A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220(2), 813–823 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  46. Tadjeran, C., Meerschaert, M.M., Scheffler, H.P.: A model for radar images and its application to adaptive digital filtering of multiplicative noise. IEEE Trans. Pattern Anal. Mach. Intell. 4, 157–166 (1982)

    Google Scholar 

  47. Teuber, T., Lang, A.: Nonlocal filters for removing multiplicative noise. In: International Conference on Scale Space and Variational Methods in Computer Vision, vol. 6667, pp. 50–61. Springer (2011)

  48. Tian, W., Zhou, H., Deng, W.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84, 1703–1727 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  49. Tur, M., Chin, K.C., Goodman, J.W.: When is speckle noise multiplicative? Appl. Opt. 21(7), 1157–1159 (1982)

    Article  Google Scholar 

  50. Ulaby, F.T., Kouyate, F., Brisco, B.: Textural information in SAR images. IEEE Geosci. Remote Sens. Soc. GE–24(2), 235–245 (1986)

    Article  Google Scholar 

  51. Ullah, A., Chen, W., Khan, M.A.: Multiplicative noise removal through fractional order TV-based model and fast numerical schemes for its approximation. Soc. Photo-opt. Instrum. Eng. (2017). https://doi.org/10.1117/12.2281822

    Article  Google Scholar 

  52. Wang, D., Gao, J.: A new method for random noise attenuation in seismic data based on anisotropic fractional-gradient operators. J. Appl. Geophys. 110, 135–143 (2014)

    Article  Google Scholar 

  53. Wang, H., Basu, T.S.: A fast finite difference method for two-dimensional space-fractional diffusion equations. SIAM J. Sci. Comput. 34(5), A2444–A2458 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  54. Wang, H., Wang, K.: An \(o(n log2n)\) alternating-direction finite difference method for two-dimensional fractional diffusion equations. J. Comput. Phys. 230(21), 7830–7839 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  55. Wang, H., Wang, K., Sircar, T.: A direct \(o(n\log _2n)\) finite difference method for fractional diffusion equations. J. Comput. Phys. 229(21), 8095–8104 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  56. Wang, Y., Liu, F., Mei, L., Anh, V.V.: A novel alternating-direction implicit spectral Galerkin method for a multi-term time-space fractional diffusion equation in three dimensions. Numer. Algorithms 86, 1572–9265 (2020)

    MathSciNet  Google Scholar 

  57. Weickert, J., Grewenig, S., Schroers, C., Bruhn, A.: Cyclic schemes for PDE-based image analysis. Int. J. Comput. Vis. 118(3), 275–299 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  58. Yang, Q., Chen, D., Zhao, T., Chen, Y.: Fractional calculus in image processing: a review. Fract. Calculus Appl. Anal. 19(5), 1222–1249 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  59. Yao, W., Guo, Z., Sun, J., Wu, B., Gao, H.: Multiplicative noise removal for texture images based on adaptive anisotropic fractional diffusion equations. SIAM J. Imaging Sci. 12(2), 839–873 (2019)

    Article  MathSciNet  Google Scholar 

  60. Yu, Y., Acton, S.T.: Speckle reducing anisotropic diffusion. IEEE Trans. Image Process. 11(11), 1260–1270 (2002)

    Article  MathSciNet  Google Scholar 

  61. Zhang, J., Chen, K.: A total fractional-order variation model for image restoration with nonhomogeneous boundary conditions and its numerical solution. SIAM J. Imaging Sci. 8(4), 2487–2518 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  62. Zhang, J., Wei, Z., Xiao, L.: Adaptive fractional-order multi-scale method for image denoising. J. Math. Imaging Vis. 43(1), 39–49 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  63. Zhang, N., Deng, W., Wu, Y.: Finite difference/element method for a two-dimensional modified fractional diffusion equation. Adv. Appl. Math. Mech. 4(4), 496–518 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  64. Zhang, W., Li, J., Yang, Y.: A fractional diffusion-wave equation with non-local regularization for image denoising. Signal Process. 103, 6–15 (2014). (Image Restoration and Enhancement: Recent Advances and Applications)

    Article  Google Scholar 

  65. Zhang, Y., Cheng, H.D., Chen, Y., Huang, J.: A novel noise removal method based on fractional anisotropic diffusion and subpixel approach. New Math. Nat. Comput. 07(01), 173–185 (2011). https://doi.org/10.1142/S1793005711001871

    Article  MATH  Google Scholar 

  66. Zhou, Z., Guo, Z., Dong, G., Sun, J., Zhang, D., Wu, B.: A doubly degenerate diffusion model based on the gray level indicator for multiplicative noise removal. IEEE Trans. Image Process. 24(1), 249–260 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (11971131, 12171123, U1637208, 51476047, 61873071), China Postdoctoral Science Foundation (2020M670893), the Natural Sciences Foundation of Heilongjiang Province (LH2021A011). The authors would like to thank Martin van Gijzen and Marielba Rojas for helping with the English and nice suggestions for the paper. The author would also like to thank reviewers and editor for the nice suggestions for the paper.

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  1. School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Xiujie Shan, Jiebao Sun, Zhichang Guo, Wenjuan Yao & Zhenyu Zhou

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  1. Xiujie Shan
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Correspondence to Wenjuan Yao.

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Appendices

Appendix A Eigenvalue estimation of matrix Q under certain conditions

In this part, we give the eigenvalue estimation of the dense matrix Q under certain conditions. The eigenvalue estimation for a sparse matrix can be derived similarly because of the absence of the non-diagonal elements. The conclusions for dense and sparse matrices are the same.

Lemma 1

For \(0<\alpha <1\), \(g_j^\alpha \) have the following properties: (1) \(g_0^\alpha =1\), \(g_1^\alpha =-\alpha <0\), \(-1 \le g_2^\alpha \le g_3^\alpha \le \cdots \le 0\), (2) \(\sum _{j=0}^\infty g_j^\alpha =0\), \(\sum _{j=0}^m g_j^\alpha \ge 0\), for any integer \(m \ge 1\).

1.1 A.1 The one-dimensional case

Theorem 1

Assume that \(c(u,|D^{\alpha }_{x+}u|)\) decreases monotonically and \(c(u,|D^{\alpha }_{x-}u|)\) increases monotonically with respect to x and \(0<\alpha <1\), then Q in (7) has M negative real eigenvalues located in the interval \([-\frac{4\alpha +4}{h^{\alpha +1}},0]\).

Proof

From the monotonicity of \(c(u,|D^{\alpha }_{x+}u|)\) and \(c(u,|D^{\alpha }_{x-}u|)\), we know that \(1 \ge c_{i-1}^+ \ge c_{i}^+ \ge 0\), \(1 \ge c_{i}^- \ge c_{i-1}^- \ge 0\) for \(i =1,\cdots ,M+1\). The sum of the absolute values of the non-diagonal entries in the ith row of Q satisfies

$$\begin{aligned} \sum \limits ^N_{k\ne i}|q_{i,k}|&= \frac{1}{h^{\alpha +1}} \sum _{k=1}^{i-1}(c_i^{x,+}g_{i+1-k}^{\alpha } - c_{i-1}^{i,+}g_{i-k}^{\alpha }) + \frac{1}{h^{\alpha +1}}c_i^{x,+}g_0^{\alpha }\\&\quad +\frac{1}{h^{\alpha +1}}c_{i-1}^{x,-}g_0^{\alpha } + \frac{1}{h^{\alpha +1}}\sum _{k=i+1}^{M}(c_{i-1}^{x,-}g_{k+1-i}^{\alpha } - c_{i}^{x,-}g_{k-i}^{\alpha })\\&= \frac{1}{h^{\alpha +1}}\left[ (c_i^{x,+}-c_{i-1}^{x,+})\sum _{k=0}^{i-1}g_k^\alpha +c_i^{x,+}g_i^\alpha - c_i^{x,+}g_1^\alpha + c_{i-1}^{x,+}g_0^\alpha \right] \\&\quad + \frac{1}{h^{\alpha +1}}\left[ (c_{i-1}^{x,-}-c_{i}^{x,-})\sum _{k=0}^{M-i}g_k^\alpha +c_{i-1}^{x,-}g_{M+1-i}^\alpha - c_{i-1}^{x,-}g_{1}^\alpha + c_{i}^{x,-}g_0^\alpha \right] \\&\quad \le -\frac{1}{h^{\alpha +1}}(c_i^{x,+}g_{1}^{\alpha } - c_{i-1}^{x,+}g_{0}^{\alpha } + c_{i-1}^{x,-}g_{1}^{\alpha } - c_{i}^{x,-}g_{0}^{\alpha })\\&= -q_{i,i}\\&\quad \le \frac{2\alpha +2}{h^{\alpha +1}}. \end{aligned}$$

Therefore, eigenvalues are of Q located in the interval \([-\frac{4\alpha +4}{h^{\alpha +1}},0]\) according to Gershgorin’s theorem. \(\square \)

1.2 A.2 The two-dimensional case

Theorem 2

For the two-dimensional case, assume that \(c(u,|D_{x+}^{\alpha }u|)\) decreases and \(c(u,|D_{x-}^{\alpha }u|)\) increases monotonically with respect to x, and \(c(u,|D_{y+}^{\alpha }u|)\) decreases and \(c(u,|D_{y-}^{\alpha }u|)\) increases monotonically with respect to y. Then the two-dimensional Q has negative real eigenvalues located in the interval \([-\frac{8\alpha +8}{h^{\alpha +1}},0]\).

Proof

From the monotonicity of \(c(u,|D^{\alpha }_{x+}u|)\), \(c(u,|D^{\alpha }_{x-}u|)\), \(c(u,|D^{\alpha }_{y+}u|)\), and \(c(u,|D^{\alpha }_{y-}u|)\), we know that

$$\begin{aligned}&1 \ge c_{i-1,j}^{x+} \ge c_{i,j}^{x+} \ge 0, ~1 \ge c_{i,j}^{x-} \ge c_{i-1,j}^{x-} \ge 0,\quad \text {for}~ i =1,\cdots ,M+1, ~\text {fixed}~ j,\\&1 \ge c_{i,j-1}^{y+} \ge c_{i,j}^{y+} \ge 0, ~1 \ge c_{i,j}^{y-} \ge c_{i,j-1}^{y-} \ge 0,\quad \text {for}~ j =1,\cdots ,N+1,~\text {fixed}~ i, \end{aligned}$$

According to Lemma 1, we have

$$\begin{aligned}&c_{i,j}^{x+}g_{i+1}^{\alpha } \ge c_{i,j}^{x+}g_{i}^{\alpha } \ge c_{i-1,j}^{x+}g_{i}^{\alpha }, \quad c_{i-1,j}^{x-}g_{i+1}^{\alpha } \ge c_{i-1,j}^{x-}g_{i}^{\alpha } \ge c_{i,j}^{x-}g_{i}^{\alpha }. \\&c_{i,j}^{y+}g_{i+1}^{\alpha } \ge c_{i,j}^{y+}g_{i}^{\alpha } \ge c_{i-1,j}^{y+}g_{i}^{\alpha }, \quad c_{i,j-1}^{y-}g_{i+1}^{\alpha } \ge c_{i,j-1}^{y+}g_{i}^{\alpha } \ge c_{i-1,j}^{y+}g_{i}^{\alpha }. \end{aligned}$$

Denote \(r_{j}^{i}\) be the sum of elements along the \([(j+1)M +i]\)th row excluding the diagonal elemnents \((Q_{j,j})_{i,i}\). Then

$$\begin{aligned} r_j^i&= \sum _{n=1,n \ne j}^{N}|(Q_{j,n})_{i,i}| + \sum _{k = 1, k \ne i}^{M}|(Q_{j,j})_{i,k}|\\&= \frac{1}{h^{\alpha + 1}}[\sum _{n = 1}^{j-1}(c_{i,j}^{y+}g_{j+1-n}^{\alpha } - c_{i,j-1}^{y+}g_{j-n}^{\alpha }) + c_{i,j-1}^{y-}g_0^{\alpha } + c_{i,j}^{y+}g_0^{\alpha }]\\&\quad + \frac{1}{h^{\alpha +1}}\sum _{n= j+1}^{N}(c_{i,j-1}^{y-}g_{n+1-j}^{\alpha } - c_{i,j}^{y-}g_{n-j}^{\alpha }) + \frac{1}{h^{\alpha +1}}\sum _{k = 1}^{i-1}(c_{i,j}^{x+}g_{i+1-k}^{\alpha } - c_{i-1,j}^{x+}g_{i-k}^{\alpha })\\&\quad + \frac{1}{h^{\alpha +1}}[c_{i-1,j}^{x-}g_0^{\alpha } + c_{i,j}^{x+}g_0^{\alpha } + \sum _{k = i+1}^{M}(c_{i-1}^{x-}g_{k+1-i}^{\alpha } - c_{i,j}^{x-}g_{k-i}^{\alpha })]\\&=\frac{1}{h^{\alpha + 1}}[(c_{i,j}^{y+} - c_{i,j-1}^{y+})\sum _{n=0}^{j-1}g_n^{\alpha } + c_{i,j}^{y+}g_j^{\alpha } - c_{i,j}^{y+}g_1^{\alpha } + c_{i,j-1}^{y-}g_0^{\alpha }\\&\quad +(c_{i,j-1}^{y-} - c_{i,j}^{y-})\sum _{n=0}^{N-j}g_n^{\alpha } + c_{i,j-1}^{y-}g_{N+1-j}-c_{i,j-1}^{y-}g_1^{\alpha } + c_{i,j}^{y-}g_0^{\alpha } \\&\quad + (c_{i,j}^{x+} - c_{i-1,j}^{x+})\sum _{k=0}^{i-1}g_k^{\alpha } + c_{i,j}^{x+}g_i^{\alpha }-c_{i,j}^{x+}g_1^{\alpha } + c_{i-1,j}^{x-}g_0^{\alpha }\\&\quad + (c_{i-1,j}^{x-} - c_{i,j}^{x-})\sum _{k=0}^{M-i}g_k^{\alpha } + c_{i-1,j}^{x,-}g_{M+1-i}^{\alpha } - c_{i-1,j}^{x-}g_1^{\alpha } + c_{i,j}^{x+}g_0^{\alpha }]\\&\le - (Q_{j,j})_{i,i}\\&\le \frac{4(\alpha + 1)}{h^{\alpha } + 1} \end{aligned}$$

By using the Gerschgorin Theorem, we can obtain that the eigenvalues of matrix Q are in \(\left[ -\frac{8(\alpha + 1)}{h^{\alpha } + 1},0\right] \). \(\square \)

Using the above estimation, it can be proved that \(\Vert e^k\Vert _{\infty }\le \Vert e^0\Vert _{\infty }\), where \(e^{k} = {\tilde{u}}^{k} - u^{k}\) and \({\tilde{u}}^k\) is the solution of the proposed model. Therefore, the stability of the explicit finite difference is proved. This stability conclusion is not only applicable to the case of dense matrices. From the proof of the eigenvalue estimation, the above conclusion is also applicable to the sparse matrix.

Appendix B Experiments of the extremum principles

We use Noisy Barbara and Noisy Fingerprint images to test the extremum principle for our model. The extreme values of the Noisy images (\(L = 10\)) are shown in Fig. 14. Then denoised results of different \(\alpha \) have been tested and the extermum values are shown in Fig. 14 by FED. We observed that when \(0<\alpha \le 2\) the extremum principle stands numerically. When \(2<\alpha \le 3\), the mimimum values of both images are equal or below the mimimum values from the initial images. In Fig. 15, maximum and minimum values of the denoised Fingerprint image are given along with iterations in EFDM. By checking how the extremum values changed with iterations, we further illustrate the conclusion about the extremum principle for different \(\alpha \). The maximum (minimum) values of the image increase (decrease) when we take more iterations with \(\alpha = 2.5\) and \(\alpha = 3\). For \(\alpha \le 2\), we have that the maximum (minimum) values decrease (increase) or keep the same, which proves the extremum principle numerically. Similar conclusions can be obtained for the cases of \(L = 1\) and \(L = 4\).

Fig. 14
figure 14

Extremum values of denoised images along the different \(\alpha \). (7.1 and 592 are the extremum values of the initial noisy Barbara image. 6.9 and 584 are the extremum values of the initial noisy fingerprint image)

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Fig. 15
figure 15

The maximum and minimum values of the denoisng results are given along the EFDM iterations

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Shan, X., Sun, J., Guo, Z. et al. Fractional-order diffusion model for multiplicative noise removal in texture-rich images and its fast explicit diffusion solving. Bit Numer Math 62, 1319–1354 (2022). https://doi.org/10.1007/s10543-022-00913-3

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  • DOI: https://doi.org/10.1007/s10543-022-00913-3

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